Finite morphisms of classical varieties

Let $k$ be an algebraically closed field and let $f\text{\hspace{0.17em}⁣}:X\to Y$ be a morphism of classical varieties over $k$ in the reduced, separated, finite-type register of Classical algebraic prevarieties, regular maps, and varieties, allowing reducible and empty varieties. Here an affine open means an open subspace isomorphic to a reduced affine algebraic set with its regular-function sheaf; it need not be irreducible or be one particular principal open. In the irreducible case this agrees with Integral classical varieties in the compatible affine-atlas register and Morphisms of classical affine varieties. The empty affine model has the zero coordinate ring and is allowed as an inverse image.

An affine open subset $U\subseteq Y$ is finite for $f$ when $f^{-1}(U)$ is affine and the pullback homomorphism of regular functions $k[U]\to k[f^{-1}(U)]$ presents the coordinate ring $k[f^{-1}(U)]$ as a finite $k[U]$-module (Affine open subsets of a classical affine variety, A classical affine variety, Affine morphisms are contravariantly equivalent to coordinate-ring homomorphisms, Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Generated submodule, cyclic and finitely generated modules, module basis and free module).

The morphism $f$ is finite when $Y$ admits a finite cover by affine open subsets $U_1,\dots ,U_n$ that are finite for $f$.

Affine-locality, recorded with the definition. The condition does not depend on the chosen cover: if some finite affine cover consists of subsets finite for $f$, then every affine open subset $U\subseteq Y$ is finite for $f$. On an affine target $\mathrm{Spm}(A)$ this is the affine-communication computation of Milne's Lemma 8.19: if $B$ is a $k$-algebra and the localisations $B_{a_1},\dots ,B_{a_n}$ are finite modules over $A_{a_1},\dots ,A_{a_n}$ for elements $a_1,\dots ,a_n$ generating the unit ideal, then finitely many of those local generators, cleared of denominators, generate $B$ as an $A$-module. The passage to an arbitrary affine open of a general $Y$ is Milne's Proposition 8.21, whose proof embeds $\mathrm{\Gamma }(f^{-1}(U),{\mathcal{O}}_X)$ into a product of finitely many affine coordinate rings and compares the canonical morphism with $\mathrm{Spm}\mathrm{\Gamma }(f^{-1}(U),{\mathcal{O}}_X)$ over the members of the cover. Consequently finiteness is affine-local on the target: it may be tested on the members of any one finite affine cover of $Y$ by affine opens.

Restriction. Finiteness is preserved by restriction to open subvarieties of the target: if $f\text{\hspace{0.17em}⁣}:X\to Y$ is finite and $V\subseteq Y$ is open, then the restriction $f^{-1}(V)\to V$ is finite, because an affine open subset of $V$ is an affine open subset of $Y$.